Abstract homotopy theory in procategories

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

T IMOTHY P ORTER

Abstract homotopy theory in procategories

Cahiers de topologie et géométrie différentielle catégoriques, tome 17, n

^o

2 (1976), p. 113-124

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Article numérisé dans le cadre du programme

ABSTRACT HOMOTOPY THEORY IN PROCATEGORIES

by Timothy

^PORTER

CAHIERS DE TOPOLOGIE

ET GEOMETRIE DIFFERENTIELLE

Vol. XVII - 2 ( 1976 )

Artin and Mazur

in [1]

initiated the use of the category ^of

prosim- plicial

^{sets as a tool in}

homotopy theory.

^Their method involved

firstly

^the

formation of the

homotopy

category ^of

simplicial

^{sets and then}

passing

^to

the

corresponding

procategory. This method proves ^to be unsuitable for cer-

tain

applications

^and

by analogy

^{with a similar} situation with

categories

^of

simplicial

spectra, it ^{seems to} be advisable to reverse the order of construc-

tion. This ^was done in

[9]

^{and it allows} one to use the

homotopy

^{limit con-}

struction of Bousfield and

Kan [3]y

^or Boardman and

Vogt ([2] and [11]).

In [9]

^no attempt ^was made to

investigate

^other

homotopy

structures within the procategory.

In this paper ^an attempt is made ^to extend a sizeable amount of « ab-

stract

homotopy theory»

^{from a}

category

^{to the}

corresponding

category ^of

proobjects.

^The

meaning

^{we attach to the term «abstract}

homotopy theory»

is that

of Quillen [10J

^{or Brown}

[4] ;

^we

only

^claim

« attempt » because

^the

actual result obtained shows

that,

^on

extending

^{an abstract}

homotopy

^theo-

ry in the manner

shown,

^{some of the structure is weakened.}

However,

^{it will}

be shown that sufficient structure remains to

give

^{a sizeable amount of elem-}

entary

homotopy theory.

In future papers

particular

^{cases will be examined}

in more detail and it will be shown

just

^{how much structure} is retained.

Other

extensions are

possible;

^{for instance see}

Hastings [8] ,

^Ed-

wards and

Hastings [6]

^{and Grossman}

[71 .

^These structures retain more

of the

original homotopy theory on C

but do not allow the same sort of ap-

plications.

Finally

^{I would like to thank} Professor Wall of

Liverpool

^{and Gavin}

Wraith of Sussex for

indirectly suggesting

^this

^approach.

1. Categories

^of

^fibrant objects.

In

[4],

Brown showed that a weaker form of

homotopy theory

^than

Quillen’s

^«model category »

structure [10] _gives

^a sufficient amount of stru- cture to allow a considerable part of

homotopy theory

^to be carried out. Ex-

plicitly

he made the

following

definitions.

Let

be a category with finite

products

^{and a final}

object e .

^As-

sume

that

has two

distinguished

classes of maps called weak

equival-

ences and

librations.

A map is called ^an

aspherical

^{or trivial}

fibration

^if

it is both ^a fibration and ^a weak

equivalence.

A

path space object

^{for an}

object

^{B is an}

object BI _together

with maps

where s is ^a weak

equivalence, ( do, d1)

^{is a fibration and}

C

will be called a category

ojjibrant objects (

^{for a}

homotopy theory)

if this structure satisfies the axioms :

( F1)

If

f,

g ^are maps ^so that

g f

^{is defined and two of}

f, g

^and

jg are

weak

equivalences,

^so is the third.

( F2 )

The

composite

^{of two} fibrations is a fibration.

Any isomorphism

is a fibration.

( F3)

^{Given a}

diagram

with v a

fibration,

^{the fibred}

product

A x B exists and the

projection

I)r.*

C

A x B -&#x3E; A is ^a fibration. If v is

aspherical,

^{so is pr .}

C

(F4)

For any

object B ,

^{there exists at least one}

path space /3

^I (not

necessarily

functorial in B

).

(F5)

^For any

object B ,

the map B -&#x3E; e is a fibration.

We refer the interested reader to Brown’s paper

[4 ]

for the detailed

development

^{of this set} of axioms. For the situation in which these axioms

will be used

here,

^it is necessary ^to alter them

slightly.

^We

replace ^(F4) by

an axiom which

requires

^{that at least one} functorial

path

space

object

^ex-

ists ;

^{we will} however continue to call this one

( F4).

It is a well known result

(

^{see for} instance Artin and Mazur

[1]

^or

Duskin [5])

that any map

f : A - B

in

pro (C)

^{can be}

replaced

up ^to iso-

morphism by

^{a «level} map)), i. e. a

proobject

^{in the} category of maps

of e .

Because of this it

helps

^{to look at functor}

categories Hom (g , e)

^{before con-}

sidering

^{the more difficult case of}

pJIQ( C).

^{We shall of} course assume that

I

is

small,

^{but no}

assumption

^{rieeds be made as to other structure}

of g.

TH EOREM 1. 1. Let

Hom (g,C)

^{be the}

category of functors from 9

^{to a cat-}

egory

o f fibrant objects

^with

functorial ^(F4).

^Then

^Hom. (g, e)

^{is a cat-}

egory

o f fibrant objects.

P RO O F . We will

specify

^the

required

^structure but will leave it to the reader

to

verify

^{that this structure works.}

- Weak

equivalences:

^Let

and

be two functors and

f: X -Y

a natural

transformation; ^f

^{is a weak}

equiva-

lence if

each f ( i ): X(i) -&#x3E;Y(i)

^{is a weak}

equivalence in(?.

- Fibrations : With

f : X - Y

as

above, ^f

^{is a fibration if each}

f( i ) :

X

( i) -&#x3E;

^Y

( i )

^{is a fibration}

in e.

- Path space

object : Denoting

^the functorial

path

space

by ( )I ,

^the

path

space of

X : I -+ e

is

COROLLARY 1. 2. The class

o f weak equivalences

ⁱⁿ

Ram (g, e)

^{admits a}

calculus

of right fractions.

P RO O F . This is a direct consequence of the Theorem

together

^with

Propos-

ition

2,

_{page 424} of

[4 ] .

The

corollary implies

^{that the} category ^of fractions formed

by

^form-

ally inverting

^{the weak}

equivalences

^{is well behaved. A}

comparison

^with

the work of

Vogt [11]

suggests ^that this

«homotopy

category » ^is

equivalent

to some category of coherent

9-indexed diagrams

^{in some sense ; we will not}

pursue this idea here.

2. Categories of cofibrant objects.

In

[4] ,

^Brown

only gives explicitly

the axioms listed in n°

1 ;

^how-

ever on page 442 ^he mentions the duals of these axioms in

Proposition

^{8 .}

Although

^{it is not}

absolutely

necessary, ^we list below the dual axioms and

state the duals of Theorem 1.1 and

Corollary

^{1.2. The reason for}

stating

these axioms for ^a « category of ’cofibrant

objects »

^{is that} any

Quillen

^model

category [10] ^yields

^{both a} category of fibrant

objects

^{and a} category ^of cofibrant

objects.

^{It is this} situation which is the most

important

^{one from}

the

point

^{of view of the}

applications.

Let

be a category with finite

coproducts

^{and an initial}

object e .

We assume

that

has two

distinguished

classes of maps called weak;

equi-

valences and

co fibrations.

A map in both classes is called a trivial

co17*-

bration.

A

cylinder object

^{for an}

object

^{B is an}

object

^{B X I}

together

^with

maps

with

o-(60+61)= v B,

^the

codiagonal

map,

60+61

^a

cofibration,

^{and cr}

a weak

equivalence.

C

will be called a

category of cofibrant objects ( for

^a

homotopy theory)

^{if this structure} satisfies the axioms :

(C1)

^as

(F1) ,

(C2)

^as

( F2 )

^{with «} f ibration »

replaced by « cofibration » ( C3)

^{Given a}

diagram

with v a

cofibration,

^the

pushout ^{A V B}

exists and

v’: A - A V B

is a co-

C C

fibration which is trivial if v is.

(C4)

^{For each}

B ,

^{there is}

at least

^a

cylinder object

^BxI

in C

( which

we will need to be functorial in B

).

( C5 )

For any

object B ,

^the

map e - B

^{is a} cofibration.

EX A MP L E S.

( a) If C

is a model category in the ^sense of

Quillen [10]

^and

C

^{consists of} the cofibrant

objects,

^then

C c

^{is a category of cofibrant ob-}

jects

ⁱⁿ the above sense.

(b) If C

is a category of fibrant

objects,

^then

eop

is a category ^{of co-} fibrant

objects

in a.natural way.

REMARK.

Although

^{we will not state or} prove any of the dual

results,

^the

theorems and

proofs

of Brown’s paper

[4] easily

^{dualize and so we shall}

feel free to use any of these dual results without

explicit proof

^{in the rem-}

ainder of this paper ^or in

sequels

^to this paper.

THEOREM 2. 1. Let

Hom. (g,C) be tbe category o f functors from I

^{to a cat-}

egory

of cofibrant objects e,

^with

functorial ^(C4).

^Then

Hom(g , C)

^{is a}

category of co f i brant objects.

PROOF. As before the bulk of the work is left to the reader. If we define weak

equivalences

^as in 1.1 and cofibrations in like _manner, the result fol- lows

easily.

COROLLARY 2.2. The class

of

^weak

equivalences

ⁱⁿ

Hom.( ^g, C)

^{admits a}

calculus of left fractions.

COROLL ARY 2.3.

If Hom (g, C) is

^the

category of functors from I

^{to a clo-}

sed model

category C,

^then

JGm( g, ee)

^{is a}

^{category o f fibrant objects}

and

Hom (g, Cc.)

^{is a}

^category o f cofibrant objects. If C= Cc =Cf’

^then

Hom. (g, C)

^{has both} structures.

RE M A R K S.

( a )

In

2.3, Cf ^{is, following Quillen’s}

^notation

^[10],

^{the full ’}

subcategory of e

determined

by

^{the fibrant}

objects of e .

(b) Although Hom (g,C)

^{is both a} category ^of fibrant

and

of cofibrant

objects,

^{this does not mean it is a}

Quillen

^model category. ^In

Quillen’s

^ax-

ioms, (M1) requires

^{a certain} interaction between fibrations and

cofibrations;

there seems no reason to suppose that fibrations and cofibrations interact in this _way in

Hom. (g,C).

It is for this reason that

categories

^{of fibrant}

and cofibrant

objects

suggest themselves ^as suitable for functor

categories.

( c )

Given more structure

in C, _Hastings

^has

equipped Hom (g, C)

^with

a full

Quillen

model category ^structure

( see [8] ).

His definitions however

are not suitable for the

applications envisaged

^{for the}

theory being

^develo-

ped

^here.

3. Procotegori es.

As was mentioned

before,

any map in ^a procategory

pta( ej

^{can be}

reindexed to

give

^{a «level} map », and ^so if

e

has a

homotopy

^{structure in-}

volving

^weak

equivalences,

etc..., it is natural ^to define

f : X - Y

to be a

weak

equivalence

ⁱⁿ

pro( C) if, by reindexing,

^one gets ^a level _map

fg:

Xi - Yi

^indexed

^by

^some

cofiltering category g ,

^{which is a weak}

^equiva-

lence in the

homotopy

^{structure of}

Hom (g, C),

^and

similarly

^{for fibrations}

and cofibrations if any. From another

point

^of

view,

^this

approach

^{to defin-}

ing

^a

homotopy theory on pro (C) might

be considered as an attempt ^to

«glue»

the various

homotopy

structures in the functor

categories Hom (g, C),

^for

cofiltering g, together

via final

( or coinitial)

functors.

Unfortunately

^this

naive

approach fails ;

^{for instance the class of weak}

equivalences

^defined

in this _way is not closed under

composition.

The obvious way is ^{to «} generates ^{an abstract}

homotopy

^{from the « ba-}

sic» material

given by

^the

Hom (g, C) .

We will thus make the

following

^de-

finitions :

- Basic weak

equivalence: ^{f: X -Y}

^{is a} basic weak

equivalence

^if

by reindexing

^{one obtains a level weak}

equivalence,

^{i. e. a weak}

equival-

ence in the

image

^{of some}

Hom(g, C) with g cofiltering. ^f

will also be called a basic weak

equivalence

^{if it is an}

isomorphism.

- Basic

fibration ( or co fibration ) : ^{f: X -}

^{Y is a} basic fibration if it

can be

replaced (by reindexing) by

^{a level fibration}

(similarly

^for

cofibration).

- Basic trivial

fibration :

^{as above but with a} level trivial fibration.

- Path

space :

^If

X: I

-&#x3E;

e

then as before

X,

^{is the}

_composite

-

Cylinder object :

^If

X : I - e,

^{then :}

- Weak

equivalence : ^{f : X}

^{-Y is a weak}

equivalence

^{if it is a com-}

posite

of basic weak

equivalences.

- Fibration : f : X -&#x3E; Y is a fibration if it is a

composite

^of basic fibra- tions and

isomorphisms.

Even with these

definitions, pro (C)

^{is not a} category of fibrant

(or cof ibrant ) objects,

^but

by

^{careful use of the}

reindexing

^results

⁽ [1] Appen- dix)

available in

pro (C) ,

^{it is}

possible

^{to recover the most}

important

^re-

sults of Part 1 of Brown’s _paper

[4] .

^{We will not} prove all the intermediate steps in the

proof

^{of these more}

general results,

^but

merely

indicate how to

adapt

^the

proofs given in [4] .

First note that there is a weak form of axiom system ^satisfied

by

the classes described above. The

important

differences are that the map

(do d 1)

^{is a basic fibration and in axiom}

^(F3)

^{we can}

^only

^claim

^that,

^if

v is a basic trivial

fibration,

^{then so} is pr

( similarly

^for

( C3 ) ) .

LEMMA

3.1. (Factorization

lemma).

1 f

^u is any

map

ⁱⁿ

pro (C),

^{then u can}

be

factored

^{u =}

p i ,

where p is ^a basic

fibration

^{and i is}

right

^{inverse to a}

basic trivial

fibration.

PROOF.

Represent

^u

by

^{a level} map and

apply

^the factorization lemma in the relevant

Hom (g, C) .

Th e definition of

ohomotopic*

^is

interesting

^{in its own}

right

^{and in-}

dicates the sort of structure that is involved here.

If

f, g : A -&#x3E; B

are two maps in

pro (C)

^then

they

^are

bomotopic

^if

there is a

« homotopy s

such that and

We will write

There is

immediately

^the

problem

^of

proving = &#x3E;&#x3E;

^is transitive.We

can

reinterprete =&#x3E;&#x3E; by noting

^{that f}

⁼

g if and

only

^if

by reindexing

^one

gets ^a

diagram

so that in

LEMMA 3.2 and th en

P ROO F.

Suppose fg ⁼ _gg

^and

_gi, ⁼ hi, ;

^{then there is a small}

cofiltering

^{cat- -}

egory with

objects pairs (i, j)

^{with i}

in g, j E g’

such that there is a dia- gram

representing

^the

diagram

(with

^the vertical maps the

respective identities)

in

pro (C).

A map from

( i , j )

^to

(i’, j’)

^{is a} map of the

corresponding diagrams.

^{If we denote this}

small

cofiltering

category

by 11

^we get ^a

prodiagram ^by assigning

^{to each}

( i , j )

^the

corresponding diagram ^*( i , j ) .

Each

diagram

thus formed

simplifies

^to

give

and on

noting ^that

^{we can}

join

^{the two}

homotopies

t ogether

^to

give

^a

homotopy

and this is the

required homotopy,

^f

^{= h .}

LEMMA

3.3 (cf.

^Brown

[4] Lemma 1). Any diagram

with i ^a basic weak

equivalence

and p ^a

fibration

^{can be imbedded}

( up

^to

isomorphism)

^{in a}

diagram

with t a

composite of

^{basic trivial}

fibrations.

P RO O F . First

apply

^{in case p is a basic fibration and then}

using

^induc-

tion on the

length

^{of the}

composite. Again reindexing

^{is used to enable each}

stage ^to be done in a functor category.

It is

only

this weaker form of Lemma 3.3 which is

needed,

^{one never}

needs the stronger form where i is a weak

equivalence.

Brown’s Lemma 2 and

Propositions

^{1 and 2 now follow}

easily using

induction on the

lengths

^of

composite required

^{in the weak}

equivalences.

As a result of

this,

Theorem 1 page

425

holds and we can form the homo- topy category

hapna C) using

(where

the limit is over the category ^of weak

equivalences

^over

A )

^{as the}

hom-set.

Rather than dwell too

long

^{on the minor}

changes

^{needed in Brown’s}

subsequent proofs

^to effect the extension to

pro (C) -basically

^the

changes

are

managed by

^careful

reindexing along

^{the lines of Lemma}

^3.2

^{above and}

induction on the

lengths

^of fibrations and weak

equivalences -

^{we will leave}

the statement and

proofs

^of intermediate results to the reader and will mere-

ly

^state the theorem which is most useful in the

applications.

We first need a definition from

[4]

_{page 432}

for C pointed.

^A

fibra-

tion sequence is a

diagram ^{F - E -}

^{B in}

C) together

^{with an action}

in

Hopro (C), ^{F X D B - F,}

^{which is}

isomorphic

^{to a}

diagram

^{and action ob-}

tained from a fibration in _pro

(C) .

TH E O R E M 3.4.

If C

^{is a}

pointed category o f fibrant objects ^{and .0}

^denotes

the

loop space functor,

^then

for

any

fibration

sequence

in

p.tu1( C) ,

^{the sequence}

is

exact (

^{in the sense}

of Quillen [10] for

q ⁼

1 ) for

any A in _pro

( e) . Dualising

^{the whole of} this section ^we get ^{the dual} result.

THEOREM 3.5.

If e

^{is a}

pointed category o f cofibrant objects ^{and E}

^den-

otes the

suspension functor,

^then

for

any

cofibration

sequence

in

pm( e)

^the sequence

is exact

(

^as

before) for

^{any B i n}

pro ( C) -

The

interesting

^more

special

^{case is}

where

is ^a

pointed Quillen

model category ^{and as before}

Cf

^{is the category} of fibrant

objects

^and

C

the category ^of cofibrant

objects

ⁱⁿ

^e.

^Theorems

^3.4

^and

^3.5 apply

ⁱⁿ

pro (Cf)

and

pro ( ee) respectively

^{and the}

^A

^{of 3.4}

^{( and}

^{B of}

^3.5)

^{can be chosen}

from amongst ^those

objects weakly equivalent

^{to some}

object

ⁱⁿ

pro (Cf)

(respectively

ⁱⁿ

pro (Cc )).

As mentioned in Section

2,

^no such axiom as

Quillen’s (M1)

^can be satisfied in

pro (C)

^even

^if ee = Cf,

^but

^{by using}

^{a stronger form of}

fibration or cofibration such ^a

development

^is

possible ^{( see} [6] or [8] ).

If,

ⁱⁿ

addition, () I

^and

() X I

^are

adjoint. on (C

^{there is some}

linking

^bet-

ween the two forms of structure

( again assuming

^for

simplicity C = Cc = (Cf).

In the cases

(i) C = Kano,

^the category ^of

pointed

^Kan

complexes,

and

(ii) C = C+(Mod-A),

^the category ^of chain

complexes

^{of A-modules}

which are bounded

below,

such a

linking

^{occurs. We will}

investigate

^{those two cases} in future papers.

Department ^of Mathematics University College CORK

Republic ^{of IRELAND}

REFERENCES

1. ARTIN, M., MAZUR, B., Etale homotopy, Lecture

Notes in

Math. 100, Springer (1969).

2. BOARDMAN, J.M., VOGT, R.M., Homotopy ^invariant algebraic structures on top-

ological spaces, Lecture Notes in Math. 347, Springer ( 1973).

3. BOUSFIELD, A.K., KAN, D.M., Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer ( 1972 ).

4. BROWN, K.S., Abstract homotopy theory ^and generalized ^sheaf cohomology, ^Trans.

A. M. S. 186 (1973), 419-458.

5. DUSKIN, J., Pro-objects, Séminaire

Heidelberg-Strasbourg

1966-67, Exposé 6, I. R. M. A. Strasbourg.

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topological

^{methods in}

homological algebra,

preprint, S. U. N. Y., Binghamton.

7. GROSSMAN, J.W., ^A homotopy theory for pro-spaces, Trans. A. M. S. 201 (1975), 161-176.

8. HASTINGS, H.M.,

Homotopy theory of

prospaces I-III, preprints, S. U. N. Y. Bin-

gh ^{am ton.}

9. PORTER, T., Stability results for topological spaces, Math. Z. 140 ( 1974), 1-21.

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11. VOGT, R.M., Homotopy ^{limits and} colimits, Math. ^Z. 134 ( 1973), 11-52.